Functional Data Analysis of Tree Data Objects

Abstract

Data analysis on non-Euclidean spaces, such as tree spaces, can be challenging. The main contribution of this paper is establishment of a connection between tree data spaces and the well developed area of Functional Data Analysis (FDA), where the data objects are curves. This connection comes through two tree representation approaches, the Dyck path representation and the branch length representation. These representations of trees in Euclidean spaces enable us to exploit the power of FDA to explore statistical properties of tree data objects. A major challenge in the analysis is the sparsity of tree branches in a sample of trees. We overcome this issue by using a tree pruning technique that focuses the analysis on important underlying population structures. This method parallels scale-space analysis in the sense that it reveals statistical properties of tree structured data over a range of scales. The effectiveness of these new approaches is demonstrated by some novel results obtained in the analysis of brain artery trees. The scale space analysis reveals a deeper relationship between structure and age. These methods are the first to find a statistically significant gender difference.

Persistent homology analysis of brain artery trees

“…These results are stronger than those in Shen et al (2014), the only other study to find a statistically significant sex difference in this data set.…”

“…This contrasts with a previous study [Bullitt et al (2010)] that correlates age with total artery length and, furthermore, the TDA correlations are independent of that earlier one (Section 3.3). TDA in our context also finds stronger sex effects than the only other study [Shen et al (2014)] to find any sex difference at all (Section 3.4).…”

“…Shen et al (2014) approach this data set using representations of planar binary trees via Dyck paths, which arise in branching processes [Harris (1952)]. The bijection represents each planar binary tree as a function of one real variable, allowing application of standard asymptotic methods when trees are viewed as random objects.…”

“…A drawback of the above approaches to tree data analysis is that they require 2-dimensional embedding of the given 3-dimensional tree structure, as noted in Aydin et al (2009), Section 2.1 and Shen et al (2014), Section 2.1. For each non-leaf node, a choice must be made as to which child node goes on the left and which goes on the right (if the tree is not binary, then an ordering of the node's children is required).…”

Bivariate smoothing of mortality surfaces with cohort and period ridges

“…Silverman & Ramsay, ; Ferraty & Vieu, ; Horváth & Kokoszka, ). More recently, the trend in functional data research has been to develop or adapt methods that take account of the specific features of the problem at hand—such as jump discontinuities in neural images (Zhu et al ) and amplitude and phase variation in nuclear magnetic resonance spectral data (Marron et al )—or that are applied to new types of data including trees (Shen et al ) and graphs (Zhu et al ). Ideas from functional data analysis are also being applied in other areas of mathematics and statistics including delay differential equations (Brunel et al ) and automated variable selection in functional linear models (Gertheiss et al ).…”